{
  "id": "2210.14741",
  "version": "v1",
  "published": "2022-10-26T14:19:47.000Z",
  "updated": "2022-10-26T14:19:47.000Z",
  "title": "Legendre symbols related to certain determinants",
  "authors": [
    "Xin-Qi Luo",
    "Zhi-Wei Sun"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime. For $b,c\\in\\mathbb Z$, Sun introduced the determinant $$D_p(b,c)=\\left|(i^2+bij+cj^2)^{p-2}\\right|_{1\\leqslant i,j \\leqslant p-1},$$ and investigated the Legendre symbol $(\\frac{D_p(b,c)}p)$. Recently Wu, She and Ni proved that $(\\frac{D_p(1,1)}p)=(\\frac {-2}p)$ if $p\\equiv2\\pmod 3$, which confirms a previous conjecture of Sun. In this paper we determine $(\\frac{D_p(1,1)}p)$ in the case $p\\equiv1\\pmod3$. Sun proved that $D_p(2,2)\\equiv0\\pmod p$ if $p\\equiv3\\pmod4$, in contrast we prove that $(\\frac{D_p(2,2)}p)=1$ if $p\\equiv1\\pmod8$, and $(\\frac{D_p(2,2)}p)=0$ if $p\\equiv5\\pmod8$. Our tools include generalized trinomial coefficients and Lucas sequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-26T14:19:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11C20",
      "11A15",
      "11B39",
      "11T99",
      "15A15"
    ],
    "keywords": [
      "legendre symbol",
      "determinant",
      "odd prime",
      "generalized trinomial coefficients",
      "lucas sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}